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Weierstrass Gap Sequence at Total Inflection Points of Nodal Plane Curves

Let $Γ$ be a plane curve of degree $d$ with $δ$ ordinary nodes and no other singularities. If $P$ is a smooth point on $Γ$ then the Weierstrass gap sequence at $P$ is considered as that at the corresponding point on the normalization of $Γ$. A smooth point $P\inΓ$ is called a total inflection point if $i(Γ,T;P)=d$ where $T$ is the tangent line to $Γ$ at $P$. There are many possible Weierstrass gap sequences at total inflection points. Our main results are: Among them (1) There exists a pair $(P,Γ)$ such that the gap sequence at $P$ is the minimal (in the sense of weight). (2) There exists a pair $(P,Γ)$ such that the gap sequence at $P$ is the maximal (resp. up to 1 maximal). And we characterize these cases in the sense of location of nodes.